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Primes and divisibility

Prove the following without using Fermat's little theorem or Euler's totient theorem. Here \(a\ |\ b\) means \(a\) divides \(b\).

  1. Prove that for every prime number \(p\) and every pair of integers \(a\) and \(b\),

    \(p\ |\ ((a+b)^p -a^p -b^p)\)

  2. Prove that for every integer \(a\) and every prime number \(p\)

    \(p\ |\ (a^p - a).\)

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